Each day, the financial markets produce a wide variety of market information that continues to evolve dynamically over the course of that day. Much of the financial information that market participants need, however, is not directly observable from the markets, but must be derived from market information through arbitrage relationships, as described below.
By way of illustration, one important example of financial information not directly observable from the markets comprises zero prices and their equivalent, zero rates. On a practical level, zero prices provide the foundations for valuing financial positions, including assets such as stocks and bonds, liabilities such as loans and other financings, and derivatives, and, more generally, for valuing future payments, whether derived from financial or other business projects, or from other sources. On a more theoretical level, zero prices provide a robust and reliable basis for identifying profitable financial transactions, and for managing the risk associated with those transactions, as discussed in, for example, D. Duffie, Dynamic Asset Pricing 31 (3rd ed Princeton University Press 2001) and J. Hull, Options, Futures, and Other Derivatives (7th ed. Prentice Hall 2008). Another example of unobservable information that market participants need comprises forward rates and forward prices, which generally comprise rates and prices that may be fixed at no cost in the market today for payment or receipt on a specified future date. Although zero and forward rates and prices are generally not directly observable from the financial markets, they may be derived from observable market information through methods comprising mathematical operations that reflect arbitrage relationships. Prior art filings in respect of market information include U.S. Pat. No. 7,734,533, U.S. Pat. No. 6,304,858, U.S. Patent Application No. 2010/0063915, U.S. Patent Application No. 2007/0061452, U.S. Patent Application No. 2007/0016509, U.S. Patent Application No. 2005/0273408 and U.S. Patent Application No. 2003/0093351.
U.S. Patent Application No. 2005/0273408 discloses systems for managing financial market information by generating a multidimensional graphical depiction of market data for at least two financial instruments. U.S. Patent Application No. 2007/0016509 discloses a computer-aided method for establishing a secondary market-relevant price for a bond issued by a corporate entity based in part on accessing data relating to executed secondary market trades for bonds issued by the entity. U.S. Patent Application No. 2007/0016509 also discloses generally a method for generating the portion of an interpolated yield curve relevant to an identified bond with a particular maturity and other specified terms. The curves drawn in the FIGS. 7 through 9 of the '509 application appear to represent simple linear interpolation, although the text of the application references cubic splines.
U.S. Patent Application No. 2007/0016509 discloses a computer aided method for establishing secondary market-relevant prices for corporate bonds which involves accessing data relating to executed secondary market trades for bonds issued by a corporate entity and using a bond pricing model to calculate an anticipated price for another bond issued by that entity. U.S. Pat. No. 6,304,858 discloses a method, system, computer program product, and data structure for trading in which a standardized contract is traded. This filing also states a known mathematical formula for bootstrapping interest rates to compute zero prices, using the resulting zero prices to value an actual or hypothetical bond, generally for purposes of establishing a contract intended to offer the contractual parties similar interest rate sensitivity to that of an interest rate swap. While this patent discloses a single formula for deriving zero prices, it does not disclose any method for obtaining market data, nor does it disclose how its formula, an older one that has several disadvantages relative to more modern methods, such as constant forward rate interpolation, would be implemented in a machine. Other examples of zero price/zero rate estimation appear in the literature, see, e.g., Ferstl & Hayden, “Zero-Coupon Yield Curve Estimation with the Package termstrc,” Journal of Statistical Software, Issue 1 Vol. 36 (August 2010), Monetary and Economic Department, “Zero-coupon yield curves: technical documentation,” 25 BIS Papers (October 2005) (available at http://www.bis.org/publ/bppdf/bispap25.pdf), but these, too, typically employ unreliable interpolation methods.
U.S. Patent Application No. 2007/0061452 discloses a system for providing notification, generally through a paging network, of specified market conditions to be monitored. U.S. Pat. No. 6,772,146 discloses a system for retrieval and display of financial information. Users may customize the types of information displayed by a portal display page. U.S. Patent Application No. 2003/0093351 discloses a fixed rate investment value modeling system and methods to calculate the current price or market value of a fixed rate investment, such as a Certificate of Deposit, based on instrument-specific data and market-specific data provided to the system. U.S. Patent Application No. 2010/0063915 similarly discloses a system and methods for providing pricing of mortgage-backed securities and other financial instruments, which determine, among other things, a continuously compounded interest rate spread corresponding to the option adjusted spread prevailing at the market.
Other patent filings in this general area relate primarily to trading. See generally the US Patent References in U.S. Pat. No. 7,734,533, disclosing a method and system for providing electronic trading via yield curves and citing extensive filings of related prior art.
Other examples of prior art are commercial information services, such as those provided by Thompson Reuters Corporation or by Bloomberg L.P. The pricing and trading systems of financial institutions, particularly the larger and more sophisticated institutions, comprise much of the prior art on the acquisition, organization, development and display of observable market information and derivation of unobservable information from that observable market information. These systems typically obtain the observable information from prices and rates on specified transactions in “benchmark” financial instruments selected to meet certain requirements, including the following. The instruments must be traded in sufficient amounts and with sufficient regularity that the associated prices or rates are realistic representatives of what can be achieved in the market. The purpose of this requirement is that there should be a “liquid” market for the instruments. In this context, the existence of a “liquid” market for an instrument means, among other things, that at the time a price of or rate on the instrument is observed on contemporaneous market transactions, a hypothetical market participant could purchase or sell a significant amount of that instrument at a price, or invest or issue a significant amount of that instrument at rate, that is reasonably close to the observed price or rate, without affecting instrument's the market price and subject to acceptably small transaction costs. Such transactions costs include the spread between purchasing and selling the instrument, referred to as the “bid-ask spread”.
In broad overview, then, many sophisticated financial institutions intend that “benchmark” financial instruments should, among other requirements, have the property that new transactions in those instruments may be executed on close to the same terms as those observed on transactions which have just been executed. In other words, the prices and rates observed on recent “benchmark” instruments should be predictive for the immediate future. As discussed below, notwithstanding this objective, finding prices and rates on observable benchmark instruments with this predictive property can be challenging for even the largest and most sophisticated institutions, particularly with regard to the implicit prices and rates not observable in the market that are derived from observable prices and rates.
Having selected a set of observable “benchmark” instruments meeting the liquidity and other requirements, sophisticated financial institutions obtain the associated prices or rates for the selected “benchmark” instruments from their own trading desks, from third party market makers or other market participants, or from external electronic data sources, and program their computers to compute zero prices and otherwise derive needed unobservable financial information using one or more of the methods known in the financial art, certain of which are described below. The derived information typically includes zero and forward rate information at certain maturities, including periodic maturities, such as semiannually, monthly or daily. A collection of rates of a particular type for every such maturity is often referred to as a “rate curve”, “yield curve” or just a “curve”, by reference to its shape when graphed. This information is then distributed or otherwise made available to specified individuals, including certain employees of the institution, through the institution's private network.
These existing systems and methods for deriving needed unobservable financial information have numerous problems, including the following. The required private systems and other infrastructure are expensive to establish and maintain, and the systems require frequent time-consuming updates and configuration. Regulatory, organizational, political and other restrictions and requirements lead to duplicate and overlapping systems, further increasing costs. Market disruptions or dislocations disrupt such systems, occasionally producing data that is flawed, misleading or useless, and not infrequently causing such systems to crash. The infrastructure that develops the curves and other information is a component in the institution's larger systems, and so must communicate with, and is dependent upon, other internal and external systems. The communication systems and methods require additional maintenance and configuration for the required interfaces and protocols, providing another source of errors. The instrument-by-instrument approach to data acquisition aggravates the problems. If data for each instrument is acquired separately, the aforementioned problems may arise separately with respect to each instrument or security “feed”, corrupting the development of information for all the instruments, since the derived information for a specified maturity frequently depends on all the observed information for all earlier maturities, as well as that for the next consecutive maturity.
These and other problems of existing private systems create high infrastructure and other costs, and also create periodic delays and inaccuracies, and insufficient reliability. Other problems include inconsistency, inaccessibility and lack of transparency. Inconsistency arises because of the instrument-by-instrument approach and the private infrastructure, which must interface with multiple internal systems. Inaccessibility arises because of the private system infrastructure required to access the data, which frequently must be further restricted within an organization for political, organizational and technological reasons. Lack of transparency arises because of organizational and practical issues surrounding private systems, and the desire for privacy. Because the associated processes are bespoke, description, disclosure and communication of these processes, all of which are required for transparency, require significant investment, which the host organization may be unwilling or unable to make. In addition, the private and potentially proprietary nature of the host systems raises further issues of confidentiality. Political and organizational concerns may also limit transparency. In the absence of transparency, the inconsistencies, inefficiencies, outright errors and other problems that systems and methods providing complex data and data processing, such as those discussed above, may develop are difficult to detect and correct. Moreover, corrections, improvements, updates and other modifications to such complex systems are made considerably more difficult by lack of transparency, since the details about the design, performance and even the objectives and history of such systems may not be readily available.
Notwithstanding their lack of transparency, reliability and other problems, these private systems are thought to achieve certain desired objectives, such as privacy, competitive advantage, timeliness and customization. As discussed elsewhere herein, the objectives of privacy and competitive advantage carry a high cost in terms of lack of transparency, and in any event are most valuable to a narrow and highly-specialized segment of market information users. Embodiments of the present invention achieve the remaining objectives almost as well, and in many cases better, than the private systems of prior art, but at far lower cost, while offering transparency and other advantages. First, with respect to privacy and competitive advantage, while certain institutions treat their curve development methodology as proprietary and seek vigorously to keep it confidential, the primary objective of curve development, including zero and forward prices and rates, is to derive accurate and timely information from current observable market levels. Observable information on current market levels is either widely known, or will be so shortly. The more reliable methods of deriving unobservable information, like zero and forward prices, from observable market information are known in the financial art. As discussed above, transparency in respect of derivation methods facilitates diagnosis and correction of problems with, and improvements, updates and modifications to, systems implementing these methods. Because these derivation methods are widely known and widely disseminated, including in the academic community, the competitive advantages of proprietary methods in private systems are modest, and relate primarily to high-frequency proprietary trading, a small community to which the present invention and its embodiments are not primarily directed.
As a result, the usefulness of privacy and confidentiality in respect of current market information, and in respect of methodology to acquire such information and derive other information from such acquired information, is modest with respect to most users of such information. By contrast, forecasting models with the objective of predicting future market levels are most valuable when their methodology is confidential, since to obtain a transaction in the market for less than its predicted future market value, the transaction's current market price must not fully reflect that future market value prediction, which is less likely if the prediction is widely known. Confidentiality with respect to future market predictions, and related methodology, is thus potentially very valuable. By making data, including stored data, available to users at low cost, embodiments of the present invention facilitate the research into, and analysis of, current and historic data that developers and users of such predictive models require.
With respect to timeliness, information on prices and rates for transactions that may be executed currently in the market is elusive even in a liquid market. Pricing on transactions that may be executed currently is sometimes referred to as “live” pricing and differs from pricing on past transactions which, however recent, are merely indicative of current market levels.
Unfortunately, the only assured route to currently available “live” prices or rates is to obtain live quotes from a buyer, seller, lender, issuer or other provider of transactions that is prepared to enter into transactions on the quoted terms. Such quotes are, however, too narrow to provide a basis for a curve of prices or rates. Providers generally will not make a broad range of live price or rate quotes available, nor will they hold them open over an extended period as terms upon which the provider feels bound to transact. For one thing, should the market move away from such terms, a provider that held such terms open may experience an economic loss were that provider to transact on those outdated terms. More generally, providers are in the business of transacting, not of providing market information. They may make a price or rate at which they will transact available to a customer for a brief period in order to facilitate a transaction with that customer, assuming the provider believes the customer is negotiating in good faith.
The alternative to live quotes is a list of the prices or rates at which recent transactions have been consummated by third parties. Such a list, because it represents dated information, akin to a car that becomes “used” the moment it is driven off a car dealers' lot, is unlikely to represent the prices and rates that new parties can actually achieve, for several reasons, among them the following. The counterparty in a previous transaction may have fully satisfied their appetite for the particular instrument, and the remaining parties may offer less attractive terms. Even in the absence of market changes, there is no assurance that such a list of executed transactions between particular parties is representative of what transactions are available to different parties, even for transactions in the same size, and even less so for transactions in different sizes.
In short, a discussion with a dealer who stands ready to transact is unlikely to be representative of a full range of available terms, although it may lead to a single transaction on a single set of terms. Such discussions are accordingly inappropriate sources for a full curve. By contrast, a broad list of actual executed transactions may not be representative of the terms a dealer who stands ready to transact will offer for new transactions, although such a list may provide a general indication of such terms. Furthermore, much of most important data is unobservable, and accordingly must be derived by a computational method from the data on such a list. The derivation methods in the financial art rely on certain assumptions, including the assumed absence of arbitrage. As discussed below, this assumption is based on the strong economic forces that arbitrage creates to exploit and eliminate it, through the possibility of a “riskless profit”. The rational for this assumption is strong, but that rational does not imply that arbitrage can never exist, only that significant arbitrage is unlikely to persist over an extended period. Moreover, certain other of the assumptions on which the derivation methods rely are based on more general considerations, such as the probable shape of certain rate curves. These assumptions are subject to less certainty than the assumed absence of arbitrage.
For all these reasons, the price and rate curves, and other financial data, derived in the prior financial art have limited predictive accuracy, notwithstanding the emphasis on using recent data to derive them. The derivation methods of prior financial art accordingly have the problems described above, including expense, unreliability, inaccuracy, inconsistency, inaccessibility and lack of transparency, and in addition offer somewhat limited value as predictions of current levels at which new transactions may be executed. As described in greater detail below, embodiments of the present invention solve these problems by achieving low cost, reliability, accuracy, consistency, accessibility and transparency, with only modest loss of predictive value.
Relative to private systems, however, the currently preferred embodiments of the present invention, by achieving transparency, necessarily sacrifice privacy to a certain extent; for example, the development and derivation methods of these embodiments are preferably fully disclosed to users and based on open-source programming tools and systems.
Users may, nonetheless, use the data and other information provided by these embodiments in any manner they see fit, including as the basis for analysis and other processes that are wholly or partially private or proprietary. Moreover, certain embodiments provide and expose one or more application programming interfaces, or “API”s, providing users means to incorporate the information these embodiments provide directly into their own private systems. Other embodiments provide users means to develop private methods that may be incorporated into the embodiments and used exclusively by those users.
As described below, by obtaining benchmark data at low cost, or no cost, from compilations, such as those provided by a public or quasi-public entity such as the Federal Reserve Bank or a financial exchange, the currently preferred embodiments eliminate the costs and many other problems associated with the private systems of the prior financial art. By using such data and standardizing derivation and development methods, these embodiments also provide consistency, reliability, accessibility and transparency. Consistency is enhanced by using data compilations, because the process of acquiring data from each such compilation need only be configured once, rather than instrument by instrument, which is difficult even where the data for all instruments comes from a single source. Changes in compilation formatting and conventions (such as, for example, quotation basis) are infrequent and are typically announced well in advance, facilitating efficient and effective updates to such configuration. Transparency is provided by using, where possible, open source, publicly available software and publicly available data, and fully disclosing all relevant development and derivation methodology, so that users generally may inspect and verify embodiments' data and processes. Similarly, reliability is enhanced by using data distributed widely, for example, by a public or quasi-public entity, and separating all component processes into individual modules, submodules, classes and subclasses that are easy to inspect and maintain. Accessibility is provided by the features just described, and the currently preferred communication methods over one or more public or private computer, telecommunication or other networks, offering access to wide communities of potential users.
Using third-party compilations for data results in a certain amount of delay for the data to be compiled and distributed. However, as a result of the Internet and other advances in telecommunications, the delay associated with distribution is minimal. The delay associated with the compilation process can be meaningful, not infrequently two days or occasionally longer, but, as discussed above, “live” pricing is in any case elusive, and reasonably contemporary data, of the sort currently preferred embodiments of the present invention generally provides, is generally sufficient to analyze potential transactions under current market conditions. Moreover, as indicated above, certain of the currently preferred embodiments of the present invention provide data entry and modification means for users to update, add to or otherwise modify the data the system provides, allowing up-to-the-minute timeliness where users feel it is needed. Certain of these embodiments provide and expose one or more APIs, allowing users to provide real-time data addition, modification and enhancement. Users who require truly live pricing, for example, in connection with transaction execution, will obtain it from transaction providers, as they do currently; embodiments of the present invention provide to such users detailed analytics to interpret and validate the quotes from such transaction providers. In particular, users of these embodiments may enter the new data provided by transaction providers and immediately derive complete rate curves using the embodiments' derivation methodology.
Based on appropriate data, certain embodiments of the present invention will also derive rate curves in one or more currencies, such as US Dollars, European Euros, British Pounds Sterling, Canadian Dollars or Japanese Yen. Certain other embodiments will derive foreign exchange rates, commodity prices, such as oil or soybean contract prices, credit spreads or equity or equity index prices, on a current and a forward basis, or, where available, futures contracts levels. Other embodiments will provide analysis and evaluation of many or most common transactions, including derivatives based on one or more of the foregoing underlying tradable assets, including forward contracts or swaps, or puts, calls or compound, complex or other options, or other derivatives, as described in greater detail in 0 below, “Note on Arbitrage-Free Pricing”.
Note on Arbitrage-Free Pricing
One important example of market information, reflected in certain of the present invention's currently preferred embodiments, is that of zero prices. Zero prices, a term familiar to those skilled in the financial art, refer to the prices of zero coupon bonds, which are bonds that provide for only a single payment at maturity and no interim coupon payments. A zero coupon bond thus has only two payments, its issue price and its redemption payment at maturity. Such a bond's yield is thus uniquely specified by (a) the quotient of the latter payment divided by the former payment, together (b) with the bond's maturity. The yield of a bond is a well-known term in the financial art that describes the annual or other periodic rate of economic income a bond provides for its holder, represented by the relationship between the bond's payments that the holder receives relative to the bond's price that its owner pays. For a zero coupon bond, the inverse of this quotient, equal to the bond's issue price divided by its redemption price, is referred to in the financial art as the zero price for that bond, and provides a simple computational method for determining the present value today of any payment on the bond's specified future maturity date made by the issuer of the bond, or other payer of the same credit quality or class. As described below, the required present value is just the nominal value of the future payment multiplied by the zero price.
In the financial art, the zero price is sometimes also referred to as a “discount factor”, reflecting the fact that the zero price is less than one as long as the bond's yield is positive, consistent with the general experience with nominal interest rates. The length of the period from a bond's issue date to its maturity date is referred to as the bond's “tenor”, although the term “maturity” is sometimes also used, when the context makes clear the reference is to the length of a period rather than to a specific maturity date. Accordingly, zero prices form a foundation for financial valuation. By establishing the price today for a fixed future payment on a specified future maturity date, with no payments prior to that maturity date, a zero price establishes the ratio between the value of the future payment today and the amount of that payment in the future. In the absence of arbitrage, as described below, this ratio must be the same for every zero coupon bond of a single issuer with that maturity date, and thus provides a procedure for determining the value today of any payment on that future date.
The value of a bond's future payments, determined by arbitrage methods as discussed below, need not be the same as the bond's market price, but if they differ, an arbitrage may arise. If arbitrage should arise, however, it is unlikely to persist in any significant size, for the following reasons. Suppose there is a zero coupon bond that pays $1000 at its maturity in ten years and sells at a price of $500 today. The zero price associated with this bond is $500/$1000, or fifty percent, equal to the ratio between the bond's current sales price and the future payment.
Were the issuer of the bond to promise to pay $2000 in ten years in a separate financial or commercial transaction, that promise must be worth $1000 today, determined as fifty percent of $2000, in the absence of arbitrage. An arbitrage transaction is one that offers the certainty or possibility of a receipt without the risk of a payment. If such a transaction becomes available, natural market forces will eliminate it. For example, suppose a bond issuer issued for $500 each a first class of bonds as above, each bond providing a single payment at maturity in ten years of $1000, and no other payments, and also issued for $1200 each a second class of bonds, each providing a single payment at maturity in ten years of $2000, and no other payments. Using short sales, a technique well-known in the financial art, a market participant may sell the latter bonds short and buy the former bonds, and thus create an arbitrage.
More specifically, a market participant may sell a bond of the second class short, raising $1200 in proceeds immediately but incurring an obligation to pay $2000 in ten years. That participant could then use $1000 of the $1200 short sale proceeds to purchase two bonds of the first class, assuring a receipt in ten years of $2000 for every two bonds purchased. The aggregate net consequences of the short sale and the purchase would be a receipt of $200 immediately per bond sold short and no net payment obligation at any time. This is an arbitrage: a receipt with no risk of any payment. Because a riskless profit is attractive, market participants would execute this transaction as much as they could. The effect of this activity by market participants would be to increase the price of the first class of bonds, by increasing demand through purchases, and decrease the price of the second class of bonds, by increasing supply through sales. The activity would cease when the price of the first class of bonds had risen and the price of the second class of bonds had fallen until the latter was exactly twice the former, since only in that event would arbitrage disappear.
Arbitrage is accordingly an unstable phenomenon, itself providing incentives for market participants to eliminate it. Moreover, the greater the riskless profit an arbitrage offers, the greater the incentives for market participants to act to eliminate it. It is therefore conventional in the financial art to assume the absence of arbitrage in analyzing market transactions. While this assumption is not invariably met, arbitrages offering significant profits rarely persist over significant periods, for the reasons described above, and financial professionals and informed private actors alike accordingly make assuming the absence of arbitrage a cornerstone of their analysis and decision-making.
As described above, a bond's yield is inversely related to the bond's price; the higher the latter, the lower the former. Absence of arbitrage may thus be expressed as the equality of the yield of the two classes of zero coupon bonds of the same issuer with the same tenor described above. In the absence of arbitrage the two classes of bonds will have the same relationship between their price and their payments, in each case consisting of a single payment at their common maturity. In the simple case of a zero coupon bond, the relationship can be expressed by the zero price, the quotient of the price divided by the payment at maturity.
More broadly, two bonds of issuers of the same credit quality with the same payment terms, up to scaling by a constant to reflect the bonds' sizes, should generally provide the same yield in the absence of arbitrage since, should the yields differ, for example, the credit risk of one can be exchanged in the derivatives market for the credit risk of the other and an arbitrage then constructed in a similar manner to that described above. Two zero coupon bonds have the same payment terms, up to scale, if they have the same maturity date and the same payment on that date, since the maturity date is the only payment date.
Each credit class of issuers thus corresponds to a consistent series of zero coupon bond yields indexed by maturity, one yield for each maturity. Such a series of yields, or rates, is often referred to as a yield curve or rate curve and is graphically represented as a curve with maturities on the x-axis and yields on the y-axis. An interesting example of such curves are the forward curves and data available for the Euro zone at http://www.ecb.europa.eu/stats/money/yc/html/index.en.html. The curves and data include zero (spot) rates, par rates and instantaneous forward rates. In the ECB's Svensson model, the par rates are derived quantities.
There is an attractive source of yield information for bonds, well-known in the financial art as the interest rate swap, or just the “swap”, market. This market has standardized terms and is very large, reported to be $342 trillion in notional amount as of June 2009. Although swaps are interest rate derivatives, not bonds, the rates in the swap market are considered to be reasonable indications of the yields on bonds with similar maturities issued by issuers with a strong credit rating, such as AA or Aa2 from Standard & Poor's and Moody's, respectively.
Swaps typically have maturities no less than one year, so for shorter maturities the source of yield information is a different type of interest rate derivative, known as Eurodollar contracts. The yields quoted on an instrument, such as a swap or Eurodollar contract, depend on several conventions, which may differ between different instruments. The yields associated with Eurodollar contracts thus need to be adjusted to make them comparable to the yields associated with swaps. Certain adjustments reflect the difference in “quotation basis”, such as “Act/360” or “30/360” day counts, or conventions for business day calendars. Other adjustments are sometimes referred to as “convexity adjustments” in the financial art; for Eurodollar contracts with maturities less than a year these convexity adjustments are generally small.
A substantially complete yield curve may thus be constructed from the yields implicit in swaps and Eurodollar contracts together, the latter generally for maturities less than one year, the former generally for maturities one year or greater. An important issue remains, one well-known in the financial art. Swap rates (the term preferred to “yield” in the financial art in respect of swaps) reflect periodic payments, typically every six months in the United States markets, and one year in the European markets. Rates that are paid periodically are typically referred to as “par rates” because, in the case of bonds, such a rate is typically set at issuance at a level that will cause a bond carrying that rate to sell at its face, or “par”, amount. A par rate thus corresponds to a market rate at origination. The more important quantities, however, are zero coupon prices and rates, which are equivalent, since the rate on a bond, including a zero coupon bond, determines its price, up to scaling for size, and conversely, as is well known in the financial art. As described above, from a practical perspective zero prices are the foundation of valuing any transaction, and from a more sophisticated perspective, they are fundamental to finding arbitrages. One result well-known in the financial art is that, subject to certain conditions, a transaction may be structured as part of an arbitrage if and only if that transaction has positive value based on appropriate zero prices. See generally Duffie, Dynamic Asset Pricing 31, cited above. Thus, many transactions, including complex ones, may be tested for arbitrage potential using zero prices.
Moreover, zero prices provide a means of computing “forward rates”, the current market's assessment of the level of future rates. The use and derivation of forward rates is part of the financial art and will be discussed in greater detail below. As part of the financial art, these results are well-known and are described in many sources, including the reference materials cited above.
Zero prices and rates may be determined from par rates, which in turn may be observed in the market, justifying the use of the term “observables” to describe such market rates. The basic and well-known technique relies on the absence of arbitrage, and may be summarized as follows. Suppose there are two observable rates, a one year zero (coupon) rate of 5%, and a two year annual par rate of 10%. The par rate is not a zero rate, because the annual par rate represents a rate paid annually, once at the end of the first year and again at the end of the second year. However, a $100 long position in the two year par rate and a $9.52 short position in the one year zero rate, offsetting the first annual payment on the long position, has the net consequences of a derived (called “synthetic” in the art) $90.48 long position in a two year zero rate paying $110 in two years, resulting in an annualized derived zero rate of 10.62% and an associated derived zero price of $90.48/$110=82.25%. This computation is well known in the financial art. If there were an observable two year zero rate, that rate must, in the absence of arbitrage, equal the 10.62% rate derived above from the one year zero rate and the two year annual par rate. Were there a difference between the observed two year zero rate and the derived two year zero rate, an arbitrage could be created by shorting the lower rate and purchasing the higher rate, where the position in the derived zero rate would be established by establishing appropriate positions in the one year zero rate and the two year annual par rate. The existence of such an arbitrage would be profitable to market participants who identify it; these market participants would then act to eliminate the underlying difference between the two zero rates and thus eliminate the arbitrage. This analysis shows that an observable one year zero rate and an observable two year annual par rate may be combined to derive a two year zero rate, even were there no observable two year zero rate. If an observable three year annual par rate is available from the market, the observable one year zero rate, the derived two year zero rate, and the observable three year annual par rate may be combined in a manner similar to that discussed above, and well known in the financial art, to determine a derived three year zero rate, regardless of whether such a zero rate is observable in the market.
This method of deriving zero rates and prices iteratively, sometimes referred to in the financial art as “bootstrapping”, is effective where there are par rates observable on liquid instruments at each needed maturity, including each date on which a periodic payment is made on a par rate. This is seldom the case, however. For example, in the United States, par rate swaps and bonds typically provide for periodic payments, referred to as coupon payments for largely historic reasons, every six months. There are, however, few if any observable par rates on instruments with maturity an odd number of semiannual periods from the issue date of the associated instrument, and those that exist are too illiquid to make effective observables for these purposes. To be effective, an observable instrument must be traded with sufficiently liquidity that a market participant may establish long or short positions in that instrument in reasonable size without affecting the market through changing the rate or incurring undue cost. By way of illustration, in the United States liquid par swaps, and thus liquid par swap rates, exist with maturities of seven and ten years, but not with maturities of seven and one-half years, eight years or other intermediate annual or semiannual maturities.
There are a number of methods, all well-known in the financial art, for addressing the absence of observable par rates at all needed maturities, methods that permit derivation of complete par, zero and forward rate curves. The first and simplest method is linear interpolation of observable par rates to determine par rates for intermediate maturities. By way of illustration, if a seven year maturity par swap rate were six percent, and a ten year maturity par swap rate were twelve percent, the linearly interpolated eight and nine year maturity par swap rates would be eight and ten percent, respectively, and linearly interpolated par swap rates for the intermediate semiannual maturities would be determined similarly. The linearly interpolated eight and one-half year swap rate would be nine percent, for example.
While simple, linear par rate interpolation has certain infirmities, including those related to hedging with the resulting zero rates, the details of which are described in the literature related to the financial art. A more complex, but superior, interpolation method is referred to in the art as the “constant forward rate method”. This method determines intermediate zero prices first, and from them then derives intermediate par rates. As with several other methods, the constant forward rate method uses “bootstrapping” and may be described by describing the iterative step to determine the zero prices for a longer maturity, once a par rate for the longer maturity and zero prices for all shorter maturities have been determined. This iterative step for the constant forward rate method may be summarized as follows in the context of zero prices previously determined for all relevant semiannual maturities not longer than seven years, and an observed ten year maturity par swap rate.
The following example determines annual rates, for simplicity. Accordingly, the observed ten year par swap rate is assumed to pay on each annual anniversary of the date accruals begin, known as the swap's “start date”, from one to ten years. First, a brief description of forward rates is helpful. Ignoring certain technicalities not material here, a one year forward rate with an effective date of seven years is a rate for one year that accrues from the seven year anniversary of the date the forward rate is specified to a maturity date, referred to as the “termination date”, one year after that seven year anniversary, which is to say the eight year anniversary of that specification date. The constant forward rate method is then as follows. Given zero prices with annual maturities of seven years or less that have previously been determined, the method picks any value (call it FR) for the one year forward rate with effective date of seven years. The constant forward rate method assumes that all three of the one year forward rates with effective dates from seven to nine years, one for each of the three intermediate annual intervals (i.e. seven to eight, eight to nine, and nine to ten years), are constant and equal to the same forward rate FR that was picked for a seven year effective date. Under this assumption, as is well known in the financial art, three zero prices may be derived from FR (the constant annual forward rate picked), one for each of the three annual maturities from eight to ten years, by repeatedly dividing each previous annual zero price by the quantity (one plus the annual forward rate) (i.e. 1+FR), starting with the seven year maturity zero price and dividing by this quantity repeatedly to derive the eight, nine and ten year maturity zero prices. As is well-known in the financial art, these must be the true zero prices if, as assumed, the annual forward rates are all constant and equal to FR. The three zero prices just computed for maturities from eight to ten years, together with the seven zero prices for maturities from one to seven years (which were by assumption previously determined), provide a complete set of ten zero prices at all annual maturities from one to ten years. These ten zero prices may be applied to compute the present value of the payments on a par bond of appropriate credit quality with a ten year maturity and ten annual coupon payments at the observed ten year par swap rate, together with the principal payment at such a bond's ten year maturity date. Because the ten year par swap rate was observed in the market, and thus represents a market par rate for such a bond, the bond must have a market price, and thus, in the absence of arbitrage, a present value, of par, equal to its face amount.
This condition that such a bond's present value must equal its par amount imposes a condition on the annual forward rate FR, and there is generally a unique forward rate that satisfies this condition, which may be determined by Newton's method or other iterative, closed-form or other procedure well known in the computational art. This method of deriving zero prices for annual maturities from eight to ten years, given zero prices for earlier maturities and an observed ten year par rate, is sometimes referred to in the financial art as the “constant forward rate” interpolation method. The par rates with annual maturities from eight to ten years may then be derived from the zero prices, because bonds with those par rates and associated maturities must have a present value of par based on those zero prices. This computation is also well-known in the financial art. Sample pseudocode illustrating the constant forward rate interpolation method in the context of one of the currently preferred embodiments of the present invention appears at E] SAMPLE CODE EXCERPTS, below.
Similar arbitrage principles apply to establish relationships between (a) “forward” prices, rates and spreads with respect to equities, commodities, foreign exchange and credit, and (b) the current market, or “spot”, levels of such prices, rates and spreads. For example, the “forward price” of an asset for delivery on a specified future settlement date is the price to which parties would agree today, based on current market levels, for delivery and payment on the future settlement date. One example is the forward price for commodities under hedging contracts into which commodity producers enter to manage the price risk of the commodities they produce. Very generally, in the absence of arbitrage (a) the forward price for any tradable asset should equal (b) the asset's current market price, plus the sum of (i) interest compounded through the forward delivery date and (ii) any costs (“carrying costs”) required to hold the asset, minus (iii) any income from the asset. By way of illustration, in the case of the commodity oil, carrying costs should include storage costs. In case of equities, income from the asset should include dividends. In the case of foreign exchange, the income from the asset may comprise the interest payments on a foreign-currency denominated bond.
In broad outline, the justification for this relationship between forward and current market prices is that, should (a) and (b) differ, market participants may obtain riskless profits by purchasing or selling short the physical asset (purchasing or selling depending on the sign of the difference), until the specified future settlement date, financing the purchase with, or investing the short sales proceeds, in interest bearing debt, while also paying any carrying costs and receiving any income.
Similar relationships, based on more complex justifications, apply to options, swaps and other derivatives on tradable assets. The absence of arbitrage thus provides a general basis for pricing many derivatives on tradable assets and related quantities such as rates. These relationships and their justification are well-known in the financial art.
As indicated above, the derived zero and other prices and rates provided by certain of the present invention's currently preferred embodiments offer valuation procedures that assume the absence of arbitrage. However, where that assumption does not hold, these procedures provide users of these embodiment means to identify arbitrage, thus providing the even greater benefit of identifying transactions offering potentially riskless profits.